MathsCBSE Class 10 · Standard · Board exam 2024, set 30/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 34(a)

Set in the CBSE Class 10 board exam of 2024, paper 30/4/1, section D, question 34(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(a) · 5 marks · Long answerCompetency-basedQuadratic Equations

A train travels a distance of 90 km at a constant speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/4/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Write the time taken at each speed using time = distance ÷ speed.

Hint 2

The difference of the two times is 30 minutes = 1/2 hour; simplify to a quadratic.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let the original speed be $x$ km/h; the increased speed is $(x + 15)$ km/h.
½ mark
Time = distance/speed, and 30 minutes = $\frac{1}{2}$ h: $\frac{90}{x} - \frac{90}{x + 15} = \frac{1}{2}$
1½ marks
$\frac{90 ( x + 15 ) - 90 x}{x ( x + 15 )} = \frac{1}{2} \Rightarrow 2700 = x^{2} + 15 x \Rightarrow x^{2} + 15 x - 2700 = 0$
1 mark
$x^{2} + 60 x - 45 x - 2700 = 0 \Rightarrow (x + 60) (x - 45) = 0$
1 mark
$x = - 60$ or $x = 45$ . Speed cannot be negative, so $x = 45$ . The original speed of the train is 45 km/h.
1 mark

Answer: 45 km/h

Where the marks are

Takes original speed x and new speed x + 15 (½ mark)
Forms 90/x − 90/(x+15) = 1/2 (1½ marks)
Reduces to x² + 15x − 2700 = 0 (1 mark)
Factorises (x + 60)(x − 45) = 0 (1 mark)
Rejects −60, speed = 45 km/h (1 mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 34(a)

Hints

Step-by-step solution

Revise the chapter

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